Differentiating powers and finding a tangent

The idea

A derivative gives the instantaneous rate of change of a function. For a positive integer power, multiply by the power and reduce the power by one. Differentiate a sum term by term; a constant has derivative zero. Evaluating the derivative at a point gives the gradient of the tangent there.

Before you begin: Use powers, substitute into a function and write a straight-line equation.

Worked example

Find the tangent to the curve at x = 2.

f (x) = x^{2} + 1

At (2, 5), the tangent and the curve meet with the same gradient, 4. The dashed guides locate the point.

Step 1

Apply the power rule to the squared term. The constant has derivative zero.

f^{'} (x) = 2 x

Step 2

Evaluate the derivative for the gradient and the original function for the point’s height.

m = f^{'} (2) = 4,

f (2) = 5

Step 3

Use point-gradient form with the point (2, 5).

y - 5 = 4 (x - 2)

Step 4

Expand and rearrange. At x = 2, this line has the same height and gradient as the curve.

y = 4 x - 3

A mistake to avoid

The derivative gives the gradient, not the y-coordinate. Use the original function to locate the point and the derivative to find the slope.

Your turn.

Find the gradient of the curve at x = 2.

g (x) = 3 x^{2} - 4 x + 1

Show the solution

g^{'} (x) = 6 x - 4,

g^{'} (2) = 8

Differentiate each term, then substitute. The constant contributes zero to the derivative.

The idea

A mistake to avoid

Your turn.

Put it into practice.

Keep exploring.